Explanation:
To find the vector parametric equation for the parabola from the origin to a point P, we need to express the position vector of P as a function of a parameter. Let's assume the coordinates of P are (x, y).
(a) Vector Parametric Equation for the Parabola: We can represent the position vector of P as a linear combination of two vectors. The first vector will point from the origin to P, and the second vector will be perpendicular to it.
Let's consider the vector from the origin to P as OP. Since P lies on the parabola, the magnitude of OP is given by the equation y = x^2. Therefore, the magnitude of OP is √(x^2 + (x^2)^2) = √(x^2 + x^4).
The unit vector in the direction of OP can be found by dividing OP by its magnitude: u = OP / √(x^2 + x^4).
To find a perpendicular vector, we can take the cross product of u with the vector (0, 0, 1) (which is perpendicular to the xy-plane). This will give us a vector perpendicular to the plane containing the parabola.
Let's call the perpendicular vector v. The cross product of u and (0, 0, 1) is given by: v = u × (0, 0, 1).
Finally, the vector parametric equation for the parabola from the origin to P is given by: r(t) = tu + tv, where t is the parameter ranging from 0 to 1.
(b) Line Integral along the Parabola: To find the line integral of a vector field F along the parabola from the origin to P, we need to evaluate the line integral: ∫(F · dr) from 0 to 1, where F is the vector field, dr is the differential vector along the parabola, and the integral is taken with respect to t.
The line integral can be calculated as follows: ∫(F · dr) = ∫(F · (du/dt) dt), where du/dt is the derivative of the unit vector u with respect to t.
Substituting r(t) = tu + tv into the above expression, we can find the line integral.
Please provide the vector field F so that I can assist you further in evaluating the line integral.